Maybe, maybe. Mine won't be controlled by a microchip that's for sure.
To be honest I couldn't say how that circuit operates, however I do know that what it is
doing is discharging a capacitor through that coil. That is perfectly possible on a simple level.
Depends a lot on what you want out.
A small capacitor will charge fast but give a low current spark.
A large capacitor will charge relatively slowly but it will make a hell of a spark.
The Tippmann is certainly using a small capacitor, it's probably that blue/turquoise rectangle on the right. Obviously not electrolytic so it's going to have a small value at that size. Maybe a few nF at a few hundred volts.
The problem is that using that kind of value capacitor
with what I have (as opposed to the custom circuit and coil Tippmann uses) doesn't actually produce a spark. Or if there is one it's invisible to eye. I've only got a spark capable of igniting a gas cloud by using a minimum of a 4.7uf capacitor (the largest "small" capacitor I have at the moment) that'll charge and fire fairly quickly, it'll never do auto but it will do semi.
Of course one thing you can do which the Tippman modules switching would probably die doing is also use a "large" capacitor - large being a flash camera sized one of 80-120uf. That makes the kind of spark which is more like a small flare because there's so much current.
But a large cap is not efficient for ignition, goes through AAA batteries fairly rapidly.
So all depends what you're after.